Step 1:

$(x-a)^2 + y^2 = a^2$

On simplifying we get,

$x^2 - 2ax +a^2+ y^2= a^2$

$x^2 + y^2 = 2ax$------(1)

Step 2:

Differentiating on both sides we get,

$2x + 2yy' = 2a$

dividing throughout by 2 we get

$x + yy' = a$-----(2)

Step 3:

Substituting equ(2) in equ(1) we get

$x^2 + y^2 = 2(x+yy')x$

On simplifying and rearranging we get

$x^2 + 2xyy' = y^2$

This is the required equation.